﻿800 .'• Dr. L. Silberstein on the 



Now take the product of any number of Z-quaternions, 

 say L u L 2 , L z &c; then we see by (II.) that all the internal 

 Q's and Q c 's as it were neutralize one another, and what is 

 left is only the Q c at the beginning and the Q at the end of 

 the whole chain. That is to say the product of any number 

 of L-quaternions is again an L-quaternion. In quite the 

 same way we see, by (II. a), that the product of any number of 

 R-quaternions is again an R-quaternion. 



Notice also that, a being any physical quaternion cov. q 

 (not necessarily that implied in L or in R), aL and Ra are 

 again physical quaternions * 9 and so are also La c and a c R, 

 namely 



aL and Ra cov. q (IV.) 



La c and a c R cov. q c . . . . (IV. a) 



Thus, the alter natinq product of any number of physical 



quaternions [ab c de c ) furnishes us either an L- or R- 



quaternion or again (biquaternions covariant with) the primary 

 physical quaternions, and never anything more f. 



One remark more before leaving this subject. Suppose 

 we are given the equation 



6X = a, 



in which a, b are cov. q. "What is the relativistic trans- 

 former of X ? To get it, write the given'equation X = 5 -1 a 

 and remember that b' 1 is cov. q . Thus the transformer of 

 X will be the same as for b c a, i. e. Q c [ ]Q. In other words, 

 X will be an .L-quaternion, 



X^-^cov.Z,. (17) 



This will, of course, be still the case if we have instead of 

 b the above differential operator D, i. e. : 



if DX = a ; then X is cov. L, . . . (V.) 



or the transformer of X is Qe[ ]Q. For D has the structure 

 of q, and the entire manipulation with the Q's is done 

 precisely as before, since Q, Q c , being constant in space and 

 time, are not exposed to D's differentiating action. Similarly 

 it is seen that 



if D c Y = a Cy then Y is cov. i?, . . (V.a) 



* Or more exactly biquaternions (in Hamilton's sense of the word) 

 transforming- like the primary physical quaternions. Cf. p. 808, infra. 



t So much as to the alternating products- And' as regards the 

 products of covariant factors, like ab, I have not, up to the present, been 

 able to make out any of their possible applications to physical subjects, 

 and shall therefore not consider them here at all. 



